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Can the projection of w onto v point in the opposite direction of v ?


Yes, as it did in that example.

Example in Three Dimensions

The diagram shows:

w = (4, 2.5, 2.5)T red vector


v = (5, 0, 3.1)Tgreen vector.


Mentally project w onto v. (Hint: remember that the length of v does not matter; only its orientation. So mentally "drop" the arrowhead of w onto the dotted line).